Large induced trees in K_r-free graphs

TL;DR

This paper establishes tight bounds on the size of the largest induced tree in K_r-free graphs, disproving a recent conjecture and improving previous results for triangle-free and larger r-free graphs.

Contribution

It provides the first tight bounds for the maximum induced tree size in K_r-free graphs, including a disproval of a recent conjecture and improved bounds for triangle-free graphs.

Findings

Connected triangle-free graphs have an induced tree of size at least √n.

For r ≥ 4, induced trees of size at least (log n)/(4 log r) exist in K_r-free graphs.

Bounds are tight up to small constants.

Abstract

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem was posed twenty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order \sqrt{n}. When r >= 4, we also show that t(G) >= (\log n)/(4 \log r) for every connected K_r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal.